The algebra of strand splitting. I: A braided version of Thompson’s group \(V\).

*(English)*Zbl 1169.20021From the introduction: We construct a braided version \(BV\) of Thompson’s group \(V\) that surjects onto \(V\). The group \(V\) is the third of three well-known groups \(F\), \(T\) and \(V\) constructed by Thompson in the 1960s that have been heavily studied since.

The group \(V\) is a subgroup of the homeomorphism group of the Cantor set \(C\). It is generated by involutions and, if the metric on \(C\) is ignored, \(V\) can be thought of as rather like a ‘Coxeter group’ of permutations of \(C\). In part II [Int. J. Algebra Comput. 16, No. 1, 203-219 (2006; Zbl 1170.20306)] we find presentations for \(BV\) and \(V\) that differ only in that the presentation for \(V\) has relations of the form \(x^2=1\) that are not present in the presentation for \(BV\). Thus \(BV\) can be thought of as an ‘Artinification’ of \(V\).

As an intermediate step in understanding the group \(BV\), we also construct a ‘larger’ group \(\widehat{BV}\) that contains (and can also be shown to be contained in) \(BV\) that is somewhat more tractable. If \(BV\) is regarded as a braided version of \(V\), then \(\widehat{BV}\) is a braided version of \(\widehat V\) that contains (and can also be shown to be contained in) \(V\) and that is also somewhat more tractable than \(V\).

The group \(\widehat V\) acts on countably many copies of the Cantor set, and another view of \(\widehat{BV}\) is that it is the group \(B_\infty\) of finitary braids on countably many strands that has been modified by allowing the strands to split and recombine. This explains the first part of the title of this paper. The group \(BV\) is the subgroup in which all splitting, braiding and recombining is confined to the first strand. The group \(BV\) is thus the ‘braid group with splitting on one strand’.

The results in the paper are geometric and algebraic descriptions of \(\widehat{BV}\) and \(BV\) that reveal their algebraic structure, a derivation of a normal form for the elements of the groups, an infinite presentation for \(\widehat{BV}\), and sketches of arguments showing that the geometric and algebraic descriptions of each group are of isomorphic groups. All that we say applies with trivial modification to \(V\), and we get a similar normal form for \(V\). This normal form for \(V\) is generally known but has never been recorded. In part II [loc. cit.], we derive finite presentations for \(\widehat{BV}\) and \(BV\), and also a new finite presentation for \(V\) that is similar to that of \(BV\).

The group \(V\) is a subgroup of the homeomorphism group of the Cantor set \(C\). It is generated by involutions and, if the metric on \(C\) is ignored, \(V\) can be thought of as rather like a ‘Coxeter group’ of permutations of \(C\). In part II [Int. J. Algebra Comput. 16, No. 1, 203-219 (2006; Zbl 1170.20306)] we find presentations for \(BV\) and \(V\) that differ only in that the presentation for \(V\) has relations of the form \(x^2=1\) that are not present in the presentation for \(BV\). Thus \(BV\) can be thought of as an ‘Artinification’ of \(V\).

As an intermediate step in understanding the group \(BV\), we also construct a ‘larger’ group \(\widehat{BV}\) that contains (and can also be shown to be contained in) \(BV\) that is somewhat more tractable. If \(BV\) is regarded as a braided version of \(V\), then \(\widehat{BV}\) is a braided version of \(\widehat V\) that contains (and can also be shown to be contained in) \(V\) and that is also somewhat more tractable than \(V\).

The group \(\widehat V\) acts on countably many copies of the Cantor set, and another view of \(\widehat{BV}\) is that it is the group \(B_\infty\) of finitary braids on countably many strands that has been modified by allowing the strands to split and recombine. This explains the first part of the title of this paper. The group \(BV\) is the subgroup in which all splitting, braiding and recombining is confined to the first strand. The group \(BV\) is thus the ‘braid group with splitting on one strand’.

The results in the paper are geometric and algebraic descriptions of \(\widehat{BV}\) and \(BV\) that reveal their algebraic structure, a derivation of a normal form for the elements of the groups, an infinite presentation for \(\widehat{BV}\), and sketches of arguments showing that the geometric and algebraic descriptions of each group are of isomorphic groups. All that we say applies with trivial modification to \(V\), and we get a similar normal form for \(V\). This normal form for \(V\) is generally known but has never been recorded. In part II [loc. cit.], we derive finite presentations for \(\widehat{BV}\) and \(BV\), and also a new finite presentation for \(V\) that is similar to that of \(BV\).

##### MSC:

20F65 | Geometric group theory |

20F05 | Generators, relations, and presentations of groups |

20F36 | Braid groups; Artin groups |

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

57S25 | Groups acting on specific manifolds |

##### Keywords:

Thompson group \(V\); normal form theorems; finite presentations; monoidal categories; braided tensor categories; braided versions##### References:

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